Given the constraints on the thread fluxes that our model needs to satisfy, in the next subsection we will show that for a set of specified non-crossing regions as shown in figure REF , a locking bit thread configuration satisfying these constraints can always be explicitly constructed. The proof of the existence of this kind of locking bit thread configuration is an improvement of bulk cell decomposition method proposed in [1]}, and it adopts the recursive method. One interesting characteristic of the constructed thread configurations is that the threads in each thread bundle is uniformly distributed at the elementary regions that the threads connect and at the RT surfaces that the threads pass through.
